C. ASA
D. AAS
The two trianglesin the picturehavetwo pairsof congruentanglesand one pair of correspondingcongruent
sides.So, the trianglecongruencepostulateyou choosemusthavetwo ’s (for the angles)and one
(for the side).You can eliminatechoices and for this reason.Nowthat you are decidingbetween
choices and , you needto identifywherethe side is locatedin relationto the givenangles.It is adjacent
to one angle,but it is not in betweenthem.Therefore,you can provecongruenceusingAAS.The correct
answeris D.
ProvingPartsCongruent
It is one thingto identifycongruencewhenall of the importantidentifyinginformationis provided,but
sometimesyou will haveto identifycongruentpartson your own.You havealreadypracticedthis in a few
differentways.Whenyou weretestingSSS congruence,you usedthe distanceformulato find the lengths
of sideson a coordinategrid. As a review, the distanceformulais shownbelow.
You can use the distanceformulawheneveryou are examiningshapeson a coordinategrid.
Whenyou werecreatingtwo-columnand flow proofs,you also usedthereflexivepropertyof congruence.
This propertystatesthat any segmentor angleis congruentto itself.Whilethis may soundobvious,it can
be very helpfulin proofs,as you saw in thoseexamples.
You may be temptedto use your ruler and protractorto checkwhethertwo trianglesare congruent.However,
this methoddoesnotnecessarilyworkbecausenot all picturesare drawnto scale.
Example 2
How couldyou prove in the diagrambelow?We can alreadysee that and. We may be able to use SSS or SAS to showthetrianglesare congruent.However, to use SSS,we wouldneed and thereis no obviousway
to provethis. Can we showthat two of the anglesare congruent?Noticethat and are
verticalangles(nonadjacentanglesmadeby the intersectionof two lines—i.e.,angleson the oppositesides
of the intersection).
The VerticalAngleTheoremstatesthat all verticalanglesare also congruent.So, this tells us that. Finally, by putting all the information together, you can confirm that